Rank-one and Quantum XOR Games

Transcription

1 Rank-one and Quantum XOR Games T. Cooney 1 M. Junge 2 C. Palazuelos 3 D. Pérez García 1 and O. Regev 4 T. Vidick 5 1 Universidad Complutense de Madrid 2 University of Illinois at Urbana Champaign 3 Instituto de Ciencias Matemáticas, CSIC, Madrid 4 CNRS, Paris, and Tel Aviv University 5 Massachusetts Institute of Technology QIP 2013, Tsinghua University, Beijing, China

2 Two Player, One Round Games Referee x a b y Alice Bob Computational Complexity Interactive proof systems Efficient proof verification PCP theorem Hardness of approximation

3 Two Player, One Round Games Referee x a b y Alice Bob Computational Complexity Interactive proof systems Efficient proof verification PCP theorem Hardness of approximation Nonlocality/Bell inequalities

4 Classical XOR Games Referee x y a b Alice Bob Classical XOR games: {a, b} {0, 1} V (a, b, x, y) = V (a b, x, y).

5 Biases of (Classical) XOR Games Bias = 2 Maximum Success Probability 1

6 Biases of (Classical) XOR Games Bias = 2 Maximum Success Probability 1 Unentangled bias ω(g) (with classical resources)

7 Biases of (Classical) XOR Games Bias = 2 Maximum Success Probability 1 Unentangled bias ω(g) (with classical resources) Maximally entangled bias ω me (G) (Players can share maximally entangled state of arbitrary dimension)

8 Biases of (Classical) XOR Games Bias = 2 Maximum Success Probability 1 Unentangled bias ω(g) (with classical resources) Maximally entangled bias ω me (G) (Players can share maximally entangled state of arbitrary dimension) Entangled bias ω (G)

9 Classical XOR Games versus Quantum XOR For all classical XOR games G, we have ω(g) ω (G) Kω(G), where 1.67 K 1.783, [CHTW04].

10 Classical XOR Games versus Quantum XOR For all classical XOR games G, we have ω(g) ω (G) Kω(G), where 1.67 K 1.783, [CHTW04]. Quantum XOR: Unbounded advantage provided by entanglement ω (T n ) = nω(t n )

11 Classical XOR Games versus Quantum XOR For all classical XOR games G, we have ω(g) ω (G) Kω(G), where 1.67 K 1.783, [CHTW04]. Quantum XOR: Unbounded advantage provided by entanglement ω (T n ) = nω(t n ) Classical XOR: Maximally entangled states are optimal resource. [CHTW04] ψ n = 1 n ii n i=1

12 Classical XOR Games versus Quantum XOR For all classical XOR games G, we have ω(g) ω (G) Kω(G), where 1.67 K 1.783, [CHTW04]. Quantum XOR: Unbounded advantage provided by entanglement ω (T n ) = nω(t n ) Classical XOR: Maximally entangled states are optimal resource. [CHTW04] ψ n = 1 n ii n i=1 Quantum XOR: Unbounded advantage over maximally entangled states ω (T n ) = nω me (T n )

13 Classical XOR Games versus Quantum XOR For Classical XOR games G, ω (G) can be efficiently computed using SDP. [CHTW04]

14 Classical XOR Games versus Quantum XOR For Classical XOR games G, ω (G) can be efficiently computed using SDP. [CHTW04] For Quantum XOR games G, ω (G) can be approximated up to a constant multiplicative factor using SDP.

15 Classical XOR Games versus Quantum XOR For Classical XOR games G, ω (G) can be efficiently computed using SDP. [CHTW04] For Quantum XOR games G, ω (G) can be approximated up to a constant multiplicative factor using SDP. [CSUU08] Classical XOR games satisfy Perfect Parallel Repetition: ω (G n ) = ω (G) n

16 Classical XOR Games versus Quantum XOR For Classical XOR games G, ω (G) can be efficiently computed using SDP. [CHTW04] For Quantum XOR games G, ω (G) can be approximated up to a constant multiplicative factor using SDP. [CSUU08] Classical XOR games satisfy Perfect Parallel Repetition: ω (G n ) = ω (G) n Quantum XOR: Unbounded Violation of Perfect Parallel Repetition ω (Cn 2 ) n 2 ω (C n ) 2

17 Quantum XOR Games Referee Alice Bob Referee prepares (known) state ψ H A H B H R and sends register A to Alice, B to Bob. Referee has private register H R.

18 Quantum XOR Games Referee Alice Bob Referee prepares (known) state ψ H A H B H R and sends register A to Alice, B to Bob. Referee has private register H R.

19 Quantum XOR Games Referee a b Alice Bob Alice and Bob share an entangled state ξ H A H B. Alice and Bob apply ±1-observables X AA = X 0 X 1, Y BB = Y 0 Y 1. Return outcomes a, b {0, 1} to Referee.

20 Quantum XOR Games Referee Alice Bob Referee measures private register, depending on parity of Alice and Bob s responses.

21 Example: T n Let ψ n be the maximally entangled state in n dimensions. T n : Alice and Bob sent one of φ 0 = 1 2 ( ψ n ) φ 1 = 1 2 ( 0 0 ψ n ) with equal probability. If φ 0, respond with answers of even parity. If φ 1, respond with answers of odd parity. Orthogonal Locally distinguishable?

22 Unbounded advantage of ω (G) over ω me (G) and ω(g) ω(t n ) = ω me (T n ) = 1 n ω (T n ) = 1

23 Unbounded advantage of ω (G) over ω me (G) and ω(g) ω(t n ) = ω me (T n ) = 1 n ω (T n ) = 1 ω (T n ) = 1 can only be achieved in limit of infinite entanglement. (T 2 LTW s coherent state exchange game.)

24 Unbounded advantage of ω (G) over ω me (G) and ω(g) ω(t n ) = ω me (T n ) = 1 n ω (T n ) = 1 ω (T n ) = 1 can only be achieved in limit of infinite entanglement. (T 2 LTW s coherent state exchange game.) Classical XOR games: maximally entangled states are optimal resource. Entanglement provides advantage of at most small constant multiplicative factor: Grothendieck/Tsirelson.

25 Example: C n Referee chooses k {1,..., n} randomly. Sends one of the two states φ 0k = 1 2 ( 0 k + k 0 ) φ 1k = 1 2 ( 0 k k 0 ) each chosen with probability 1 2 to Alice and Bob. If φ 0k, respond with answers of even parity. If φ 1k, respond with answers of odd parity.

26 Large Violation of Perfect Parallel Repetition Suppose Alice and Bob play two games simultaneously and must win both sub-games in order to win. For classical XOR games, we have ω (G G) = ω (G) 2.

27 Large Violation of Perfect Parallel Repetition Suppose Alice and Bob play two games simultaneously and must win both sub-games in order to win. For classical XOR games, we have ω (G G) = ω (G) 2. However for Rank-one Quantum & Quantum XOR games: ω (C n ) = 1 n ω (C n C n ) 1 2n (ω (C n )) 2

28 Algorithms Theorem There exists a polynomial-time algorithm which, given as input an explicit description of a quantum XOR game G, outputs two numbers ω nc (G) and ω os (G) such that ω(g) ω me (G) ω nc (G) 2 2ω(G), ω (G) ω os (G) 2ω (G).

29 Techniques Theorem (Grothendieck s Inequality) Suppose that s i and t j are real numbers such that s i, t j 1. Suppose that a ij are real numbers such that a ij s i t j 1. Then i,j a ij ξ i η j ij k, for all vectors ξ i, η j in the unit ball of a real Hilbert space H. It is known that 1.67 k From this it follows that for a classical XOR game, ω (G) kω(g).

30 Techniques Theorem (Grothendieck s Inequality) Suppose that s i and t j are real numbers such that s i, t j 1. Suppose that a ij are real numbers such that a ij s i t j 1. Then i,j a ij ξ i η j ij k, for all vectors ξ i, η j in the unit ball of a real Hilbert space H. It is known that 1.67 k Noncommutative and Operator-space extensions of Grothendieck s inequality allow us to relate biases of Quantum XOR games to SDP s.

31 Quantum Games Referee Alice Bob Referee prepares (known) state ψ H A H B H R and sends register A to Alice, B to Bob.

32 Quantum Games Referee Alice Bob Alice and Bob share an entangled state ξ H A H B. Alice and Bob apply arbitrary local unitaries U AA, V BB and then send registers A and B back to referee.

33 Quantum Games Referee Alice Bob Referee performs measurement with projective measurements: {P ACCEPT, P REJECT = Id P ACCEPT }

34 Rank-one Quantum Games Referee Alice Bob Referee performs measurement with projective measurements: {P ACCEPT = γ γ, P REJECT = Id P ACCEPT } Maximum Success Probability = ω 1 (G)

35 Rank-one Quantum Games Quantum XOR Games To each Quantum XOR Game G, one can associate a Rank-one Quantum Game G such that (ω (G)) 2 = ω 1(G) To each Rank-one Quantum Game G, one can associate a Quantum XOR Game G such that (ω (G )) 2 = ω 1(G ) Thus the previous results about SDP s and Parallel repetition can be phrased in terms of either Rank-one Quantum Games or Quantum XOR games.

36 Summary 1 Classical: ω (G) Kω(G) Quantum: Unbounded advantage ω (T n ) = nω(t n ) Classical: Maximally entangled state is optimal resource. Quantum: Unbounded advantage ω (T n ) = nω me (T n ) Classical: ω (G) can be computed using SDP Quantum: ω (G) can be approximated up to constant factor using SDP Classical: Satisfies Perfect Parallel Repetition: ω (G 2 ) = ω (G) 2 Quantum: Unbounded Violation of Perfect Parallel Repetition ω (Cn 2 ) n 2 ω (C n ) 2

37 Summary 2 Generalization of classical XOR games using quantum messages.

38 Summary 2 Generalization of classical XOR games using quantum messages. Rich class of games that displays properties of entanglement not seen in classical case.

39 Summary 2 Generalization of classical XOR games using quantum messages. Rich class of games that displays properties of entanglement not seen in classical case. Remain tractable with efficient approximation algorithms for biases.

40 Summary 2 Generalization of classical XOR games using quantum messages. Rich class of games that displays properties of entanglement not seen in classical case. Remain tractable with efficient approximation algorithms for biases. Application of deep generalizations of Grothendieck s Inequality to problems in quantum information theory.

41 Summary 2 Generalization of classical XOR games using quantum messages. Rich class of games that displays properties of entanglement not seen in classical case. Remain tractable with efficient approximation algorithms for biases. Application of deep generalizations of Grothendieck s Inequality to problems in quantum information theory. Operator space theory provides both examples and techniques for studying these quantum games.

42 Thank You! Rank-one Quantum Games, arxiv: Quantum XOR Games, arxiv: Research partially supported by: Spanish grants QUITEMAD, I-MATH, MTM , S2009/ESP-1594 European project QUEVADIS NSF DMS ERC Starting Grant NSF DMS